Fractal dimension evolution and spatial replacement dynamics of urban growth

TL;DR

This paper models urban fractal dimension growth using logistic functions, revealing how spatial replacement dynamics can lead to bifurcation and chaos in city development.

Contribution

It introduces a novel framework linking fractal dimension evolution with chaos theory through spatial replacement dynamics in urban growth.

Findings

Fractal dimension evolution follows a logistic curve.

High rates of spatial replacement induce chaos.

Bifurcation occurs in urban growth dynamics.

Abstract

This paper presents a new perspective of looking at the relation between fractals and chaos by means of cities. Especially, a principle of space filling and spatial replacement is proposed to explain the fractal dimension of urban form. The fractal dimension evolution of urban growth can be empirically modeled with Boltzmann's equation. For the normalized data, Boltzmann's equation is equivalent to the logistic function. The logistic equation can be transformed into the well-known 1-dimensional logistic map, which is based on a 2-dimensional map suggesting spatial replacement dynamics of city development. The 2-dimensional recurrence relations can be employed to generate the nonlinear dynamical behaviors such as bifurcation and chaos. A discovery is made that, for the fractal dimension growth following the logistic curve, the normalized dimension value is the ratio of space filling. If the rate of spatial replacement (urban growth) is too high, the periodic oscillations and chaos will arise, and the city system will fall into disorder. The spatial replacement dynamics can be extended to general replacement dynamics, and bifurcation and chaos seem to be related with some kind of replacement process.